An engquist-osher-type scheme for conservation laws with discontinuous flux adapted to flux connections

SIAM Journal on Numerical Analysis

Volumn 47, Issue 3, 2009, Pages 1684-1712

  Bürger, Raimund ^a   Karlsen, Kenneth H ^b   Towers, John D ^c  

^a UNIVERSITY OF CONCEPCIÓN   (Chile)

^b UNIVERSITY OF OSLO   (Norway)

^c MiraCosta College   (United States)

Author keywords

Adapted entropy; Conservation law with discontinuous flux; Engquist Osher scheme; Entropy solution of type (A, B); Flux connection

Indexed keywords

ADAPTED ENTROPY; COMPENSATED COMPACTNESS; CONSERVATION LAW; CONSERVATION LAW WITH DISCONTINUOUS FLUX; DIFFERENCE SCHEMES; DISCONTINUOUS FLUXES; ENGQUIST-OSHER SCHEME; ENTROPY INEQUALITY; ENTROPY SOLUTION; ENTROPY SOLUTION OF TYPE (A, B); HEAVISIDE FUNCTIONS; NONLINEAR FUNCTIONS; NUMERICAL EXAMPLE; PIECEWISE SMOOTH; RIEMANN SOLVER; SCALAR CONSERVATION LAWS; SEMIGROUPS; SPATIAL VARIATIONS; STEADY STATE SOLUTION; TYPE-ENTROPY;

ENTROPY; SHRINKAGE;

LAWS AND LEGISLATION;

EID: 65949091995     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/07069314X     Document Type: Article

References (45)

1. ADIMURTHI, J. JAFFRÉ, AND G.D. VEERAPPA GOWDA, Godunov-type methods for conservation laws with a flux function discontinuous in space, SIAM J. Numer. Anal., 42 (2004), pp. 179-208.
2. ADIMURTHI, S. MISHRA, AND G.D. VEERAPPA GOWDA, Optimal entropy solutions for conservation laws with discontinuous flux functions, J. Hyperbolic Differ. Equ., 2 (2005), pp. 783-837.
3. ADIMURTHI, S. MISHRA, AND G.D. VEERAPPA GOWDA, Existence and stability of entropy solutions for a conservation law with discontinuous non-convex fluxes, Netw. Heterog. Media, 2 (2007), pp. 127-157.
4. E. AUDUSSE AND B. PERTHAME, Uniqueness for scalar conservation laws with discontinuous flux via adapted entropies, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), pp. 253-265.
5. F. BACHMANN AND J. VOVELLE, Existence and uniqueness of entropy solution of scalar conservation laws with a flux function involving discontinuous coeficients, Comm. Partial differential Equations, 31 (2006), pp. 371-395.
6. P. BAITI AND H. JENSSEN, Well-posedness for a class of 2× 2 conservation laws with L8 data, J. differential Equations, 140 (1997), pp. 161-185.
7. R. BÜRGER, A. GARCÍA, K.H. KARLSEN, AND J.D. TOWERS, A family of schemes for kinematic flows with discontinuous flux, J. Engrg. Math., 60 (2008), pp. 387-425.
8. R. BÜRGER, A. GARCÍA, K.H. KARLSEN, AND J.D. TOWERS, difference schemes, entropy solutions, and speedup impulse for an inhomogeneous kinematic traffic flow model, Netw. Heterog. Media, 3 (2008), pp. 1-41.
9. R. BÜRGER AND K.H. KARLSEN, On a diffusively corrected kinematic-wave traffic flow model with changing road surface conditions, Math. Models Methods Appl. Sci., 13 (2003), pp. 1767-1799.
10. R. BÜRGER, K.H. KARLSEN, S. MISHRA, AND J.D. Towers, On conservation laws with discontinuous flux, in Trends in Applications of Mathematics to Mechanics, Y. Wang and K. Hutter, eds., Shaker Verlag, Aachen, 2005, pp. 75-84.
11. R. BÜRGER, K.H. KARLSEN, N.H. RISEBRO, AND J.D. Towers, On a model for continuous sedimentation in vessels with discontinuously varying cross-sectional area, in Hyperbolic Problems: Theory, Numerics, Applications, T.Y. Hou and E. Tadmor, eds., Springer-Verlag, Berlin, 2003, pp. 397-406.
12. R. BÜRGER, K.H. KARLSEN, N.H. RISEBRO, AND J.D. TOWERS, Well-posedness in BVt and convergence of a difference scheme for continuous sedimentation in ideal clarifier-thickener units, Numer. Math., 97 (2004), pp. 25-65.
13. R. BÜRGER, K.H. KARLSEN, AND J.D. TOWERS, A model of continuous sedimentation of fiocculated suspensions in clarifier-thickener units, SIAM J. Appl. Math., 65 (2005), pp. 882-940.
14. G.-Q. CHEN, N. EVEN, AND C. KLINGENBERG, Entropy solutions to conservation laws with discontinuous fluxes via microscopic interacting particle systems, in Stochastic Analysis and Partial differential Equations, Contemp. Math. 429, Amer. Math. Soc., Providence, RI, 2007, pp. 63-76.
15. M.G. CRANDALL AND A. MAJDA, Monotone difference approximations for scalar conservation laws, Math. Comp., 34 (1980), pp. 1-21.
16. M.G. CRANDALL AND L. TARTAR, Some relations between nonexpansive and order preserving mappings, Proc. Amer. Math. Soc., 78 (1980), pp. 385-390.
17. S. DIEHL, On scalar conservation laws with point source and discontinuous flux function, SIAM J. Math. Anal., 26 (1995), pp. 1425-1451.
18. S. DIEHL, Scalar conservation laws with discontinuous flux function. I. The viscous profile condition, Comm. Math. Phys., 176 (1996), pp. 23-44.
19. B. ENGQUIST AND S. OSHER, One-sided difference approximations for nonlinear conservation laws, Math. Comp., 36 (1981), pp. 321-351.
20. M. GARAVELLO, R. NATALINI, B. PICCOLI, AND A. TERRACINA, Conservation laws with discontinuous flux, Netw. Heterog. Media, 2 (2007), pp. 159-179.
21. T. GIMSE AND N.H. RISEBRO, Solution of the Cauchy problem for a conservation law with a discontinuous flux function, SIAM J. Math. Anal., 23 (1992), pp. 635-648.
22. J.M.-K. HONG, Part I: An Extension of the Riemann Problems and Glimm's Method to General Systems of Conservation Laws with Source Terms. Part II: A Total Variation Bound on the Conserved Quantities for a Generic Resonant Nonlinear Balance Laws, Ph.D. thesis, University of California, Davis, 2000.
23. K.H. KARLSEN, M. RASCLE, AND E. TADMOR, On the existence and compactness of a twodimensional resonant system of conservation laws, Commun. Math. Sci., 5 (2007), pp. 253-265.
24. K.H. KARLSEN, N.H. RISEBRO, AND J.D. TOWERS, On a nonlinear degenerate parabolic transport-diffusion equation with a discontinuous coeficient, Electron. J. differential Equations, 2002 (2002), pp. 1-23.
25. K.H. KARLSEN, N.H. RISEBRO, AND J.D. TOWERS, On an upwind difference scheme for degenerate parabolic convection-diffusion equations with a discontinuous coeficient, IMA J. Numer. Anal., 22 (2002), pp. 623-644.
26. K.H. KARLSEN AND J.D. TOWERS, Convergence of the Lax-Friedrichs scheme and stability for conservation laws with a discontinuous space-time dependent flux, Chinese Ann. Math. Ser. B, 25B (2004), pp. 287-318.
27. K.H. KARLSEN, N.H. RISEBRO, AND J.D. Towers, L1 stability for entropy solutions of nonlinear degenerate parabolic convection-diffusion equations with discontinuous coeficients, Skr. K. Nor. Vid. Selsk., 3 (2003), pp. 1-49.
28. E.F. KAASSCHIETER, Solving the Buckley-Leverett equation with gravity in a heterogeneous porous medium, Comput. Geosci., 3 (1999), pp. 23-48.
29. C. KLINGENBERG AND N.H. RISEBRO, Convex conservation laws with discontinuous coeficients, existence, uniqueness and asymptotic behavior, Comm. Partial differential Equations, 20 (1995), pp. 1959-1990.
30. C. KLINGENBERG AND N.H. RISEBRO, Stability of a resonant system of conservation laws modeling polymer flow with gravitation, J. differential Equations, 170 (2001), pp. 344-380.
31. S.N. KRUŽKOV, First order quasi-linear equations in several independent variables, Math. USSR Sb., 10 (1970), pp. 217-243.
32. L.W. LIN, B.J. TEMPLE, AND J.H. WANG, A comparison of convergence rates for Godunov's method and Glimm's method in resonant nonlinear systems of conservation laws, SIAM J. Numer. Anal., 32 (1995), pp. 824-840.
33. L.W. LIN, B. TEMPLE, AND J.H. WANG, Suppression of oscillations in Godunov's method for a resonant non-strictly hyperbolic system, SIAM J. Numer. Anal., 32 (1995), pp. 841-864.
34. W.K. LYONS, Conservation laws with sharp inhomogeneities, Quart. Appl. Math., 40 (1982/83), pp. 385-393.
35. S. MOCHON, An analysis of the traffic on highways with changing surface conditions, Math. Model., 9 (1987), pp. 1-11.
36. D.N. OSTROV, Viscosity solutions and convergence of monotone schemes for synthetic aperture radar shape-from-shading equations with discontinuous intensities, SIAM J. Appl. Math., 59 (1999), pp. 2060-2085.
37. D.N. OSTROV, Solutions of Hamilton-Jacobi equations and scalar conservation laws with discontinuous space-time dependence, J. differential Equations, 182 (2002), pp. 51-77.
38. E.Y. PANOV, Existence of strong traces for generalized solutions of multidimensional scalar conservation laws, J. Hyperbolic Differ. Equ., 2 (2005), pp. 885-908.
39. E.Y. PANOV, Existence of strong traces for quasi-solutions of multidimensional scalar conservation laws, J. Hyperbolic Differ. Equ., 4 (2007), pp. 729-770.
40. E.Y. PANOV, Existence and strong pre-compactness properties for entropy solutions of a firstorder quasilinear equation with discontinuous flux, Arch. Ration. Mech. Anal., to appear.
41. N. SEGUIN AND J. Vovelle, Analysis and approximation of a scalar conservation law with a flux function with discontinuous coeficients, Math. Models Methods Appl. Sci., 13 (2003), pp. 221-257.
42. E. TADMOR, Numerical viscosity and the entropy condition for conservative difference schemes, Math. Comp., 43 (1984), pp. 369-381.
43. J.D. TOWERS, Convergence of a difference scheme for conservation laws with a discontinuous flux, SIAM J. Numer. Anal., 38 (2000), pp. 681-698.
44. J.D. TOWERS, A difference scheme for conservation laws with a discontinuous flux: The nonconvex case, SIAM J. Numer. Anal., 39 (2001), pp. 1197-1218.
45. A. VASSEUR, Strong traces of multidimensional scalar conservation laws, Arch. Ration. Mech. Anal., 160 (2001), pp. 181-193.